I also use Dropbox for sites and it works really well.
But be aware that Freeway will, in theory, only open on one computer at a time. I say in theory because sometimes my FW will open on my desktop and my laptop at the same time and at other times it gives me the message saying this is not allowed. There are even times when having quit FW on one computer the other one still insists that it is open and I have to restart to stop the two telling fibs to each other. Wether this would happen once the laptop is out side of my local network I do not know but I would be miffed to go somewhere and find Freeway unwilling to show its face. Have others found this happens?
This is an answer to your lead topic. Install Dropbox. This will set up a folder on your computer that will sync with every computer you use the same Dropbox account on.
Put your website in the Dropbox folder on the first system, take your second system, install the same Dropbox account and your website automatically will be on that system too. Leave it in the Dropbox folder and work on your project like you are used to.
If you go back to the first system immediately after booting the Dropbox folder will be synced and you will have the most recent version on both systems.
After reading your post again, maybe this is a better answer. Yes, you can pick up the folder from one computer and put it on the other computer and continue working. Just make sure you have all the linked files with you.
The easiest way to gather all your files is to choose save archive in Freeway.
Yes but do File>Save Archive first to make sure you have all the media in the folder then copy the folder onto a USB key or grab it from your desktop machine if you are on shared network.